Picard groups for tropical toric schemes

TL;DR

This paper proves that the Picard group of an irreducible monoid scheme remains unchanged under scalar extension to any idempotent semifield or field, linking algebraic and tropical geometric structures.

Contribution

It establishes the invariance of Picard groups under scalar extension for monoid schemes, connecting classical and tropical geometry through sheaf cohomology and divisor class groups.

Findings

Picard groups are stable under scalar extension to semifields and fields.

Picard groups of tropical schemes match those of classical toric varieties.

The group of Cartier divisors modulo principal divisors is isomorphic to the Picard group.

Abstract

From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group $CaCl(X_S)$ of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme $X_S$ and prove that $CaCl(X_S)$ is isomorphic to $Pic(X_S)$.